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<p>Test MJ scaling and font: <i>u$u$i$i$f$f$</i></p>

<p>Test ümlauts, first in the current font (normal, umlaut), then using 
in mathmode normal, using <code>\ddot</code> and <code>\"</code>:
<i>
  uü$u\ddot u\"u$,
  aä$a\ddot a\"a$,
  oö$o\ddot o\"o$,
  aå$a\phantom{a}\r{a}$
</i></p>

<a href="http://www.gnu.org">GNU project <span class="referenceNbr">[3]</span></a>

<div class="theorem" id="ttt">
  <span class="theorem"><a class="addTxtForbidden" href="#ttt">Theorem A.</a></span>
  <span>a span</span>
</div>
<div class="theorem">
  <span class="theorem">Theorem 0.1.</span>
  <p>a paragraph</p>
</div>
<div class="theorem">
  <span class="theorem">Theorem.</span>
  No enclosing tags!<p>a paragraph</p>
</div>
<div class="theorem">
  <span class="theorem">Theorem B.</span>
  <span>Enclosed in span.</span><p>a paragraph</p>
</div>
<div class="theorem">
  <span class="theorem">Theorem.</span>
  <span></span>
  <p>an empty span before this &lt;p&gt;</p>
</div>
<div class="theorem">
  <span class="theorem">Theorem 0.2.</span>
</div>
<div class="lemma">
  <span class="lemma">Lemma.</span>
  The thm above is completely empty
</div>
<div class="corollary">
  <span class="corollary">Corollary.</span>
  <span>span</span>
</div>
<div class="proposition">
  <span class="proposition">Proposition 0.3.</span>
  <p>paragraph</p>
</div>
<div class="conjecture">
  <span class="conjecture">Conjecture.</span>
  <p>paragraph</p>
</div>
<div class="definition">
  <span class="definition">Definition 0.4.</span>
  <p>paragraph</p>
</div>
<div class="remark">
  <span class="remark">Remark.</span>
  <p>paragraph</p>
</div>
<div class="proof">
  <span class="proof">Proof.</span>
  <p>
    These three sentences are a paragraph. According to <a 
    href="#ttt">Theorem A</a> we have it. Or simply by <a href="#ttt">A</a>
  </p>
</div>

<h1>Basics of the Hermite functions and transform</h1>

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<div class="motivation">
<span class="motivation">Motivation.</span>
<p>
  The reason to have this CRAZY document is just to try all the 
  different HTML, CSS and JavaScript features.
</p>
</div>

<p>
  Here is a try for citing an offline reference: <a 
  href="#articleTry"><span class="referenceNbr">[1]</span></a>, and with 
  something more descriptive than a simple number: Blåsten, 
  Meikäläinen<a href="#articleTry"><span class="referenceNr"> 
  [1]</span></a>. In particular see <a href="#articleTry"><span 
  class="referenceNbr"> [1, Chapter II]</span></a>. Note that above 
  <q>Blåsten, Meikäläinen</q> is currently not part of the anchor. If it 
  was, then it would display like <q>Chapter II</q> above.
</p>


<section>
<h2>Hermite functions</h2>

<p>
  Lorem ipsum dolor sit amet, consectetur adipiscing elit. Morbi ligula 
  lorem, rhoncus vel hendrerit a, varius a turpis. Praesent in mauris 
  non ex vehicula vestibulum sit amet in nulla. Morbi elementum justo 
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  venenatis magna. Maecenas est diam, laoreet eu porta ut, vulputate nec 
  purus.
</p>
<p>
  Vestibulum at iaculis risus. Etiam consectetur ornare ligula quis 
  tincidunt. Nulla ultricies felis a posuere interdum. Quisque interdum 
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  ornare enim pulvinar semper suscipit. Mauris facilisis nisl ligula, 
  eget tempus nisl egestas eget. Aliquam erat volutpat. Aliquam aliquet 
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  cursus mi efficitur. Aenean vel fringilla justo, nec dignissim justo.
</p>

<p>
  An unordered list:
  <ul>
    <li>asdf</li>
    <li>faeifs</li>
    <li>fewafwa</li>
  </ul>
</p>
<p>
  An ordered list:
  <ol>
    <li>asdf</li>
    <li>faeifs</li>
    <li>fewafwa</li>
  </ol>
</p>
<p>
  A description list:
  <dl>
    <dt>aasdaasdadssdf</dt>
      <dd>adaw dad wad wad wdwa daw dwaw dad wa</dd>
    <dt>faeidawdadsfs</dt>
      <dd>lijfes s fesi fhslkjadf wijasd<dd>
    <dt>fewafwadwwda</dt>
      <dd>fdwalidaw lidaw ijalw dwa ldwijald ja</dd>
    <dt>adaliwjd</dt>
    <dt>ijwefoiewf</dt>
</dl>
</p>

<p>
  The following theorem, <a href="#basics">Theorem 1.1</a>, is of utmost 
  importance. But don't forget the other stuff in the second <a 
  href="example2.html">file<span class="referenceNbr"> [2]</span></a>. 
  Namely <a href="example2.html#corolla">AA<span class="referenceNbr"> 
  [2, #corolla]</span></a> and for example <a 
  href="example2.html#eq1">AA<span class="referenceNbr"> [2, 
  #eq1]</span></a>, <a href="example2.html#cEquation">AA<span 
  class="referenceNbr"> [2, #cEquation]</span></a> and <a 
  href="example2.html#thm1">AA<span class="referenceNbr"> [2, 
  #thm1]</span></a>. Here is a link pretending to be in this text: <a 
  href="#nothere">link</a>.
</p>

<div class="theorem" title="Basic identities" id="basics">
<span class="theorem">
  <a href="#basics">Theorem 1.1 <span class="theoremName">(Basic identities)</span>.</a>
</span>
<p>
  Let $\psi_\alpha$, $\alpha\in\N$, be Hermite functions. Then
  
  \[
    \psi_\alpha'(x) = \sqrt{\frac{\alpha}{2}}\psi_{\alpha-1}(x) - 
    \sqrt{\frac{\alpha+1}{2}}\psi_{\alpha+1}(x)
  \]
  
  and
  
  \[
    x\psi_\alpha(x) = \sqrt{\frac{\alpha}{2}}\psi_{\alpha-1}(x) + 
    \sqrt{\frac{\alpha+1}{2}}\psi_{\alpha+1}(x).
  \]
  
  Moreover the Hermite functions are eigenfunctions to the quantum 
  mechanical oscillator
  
  \begin{equation}
    \label{eigenFunction}
    (x^2 - \partial_x^2) \psi_\alpha(x) = (2\alpha + 1)\psi_\alpha(x).
  \end{equation}
</p>
</div>
</section>


<section>
<h2>Hermite transform</h2>

<p>
  In this section we will first show that the Hermite functions form an 
  orthonormal set in $L^2(\R)$. After that we will show that the set is 
  complete and that leads to the Hermite transform.
</p>

<div class="lemma" id="orthonormalSequenceLemma">
<span class="lemma">
  <a href="#orthonormalSequenceLemma">Lemma 2.1.</a>
</span>
<p>
  Let $\psi_\alpha:\R\to\R$ be Hermite functions. Then
  
  \[
    \int_{-\infty}^\infty \psi_\alpha(x)\psi_\beta(x) dx = 
    \delta_{\alpha\beta}
  \]
  
  i.e. the sequence $(\psi_\alpha)_{\alpha=0}^\infty$ is orthonormal in 
  $L^2(\R)$.
</p>
</div>
<div class="proof">
  <span class="proof">Proof.</span>
  <p>
    Note first the identity $(x+\partial_x)(x-\partial_x) = 
    (x^2-\partial_x^2) + 1$. Combine it with the fact that $\psi_n$ is 
    an eigenfunction for the quantum oscillator \eqref{eigenFunction} to 
    get
    
    \[
      (x+\partial_x)(x-\partial_x)\psi_n = 2(n+1)\psi_n
    \]
    
    for any $n\in\N$. Note also that the transpose of $x-\partial_x$ in 
    the $L^2$ inner product is $x+\partial_x$.
  </p>
  <p>
    Hence we get
    
    \begin{align*}
      &\int \psi_\alpha \psi_\beta dx = \frac{1}{2\sqrt{\alpha\beta}} 
      \int (x-\partial_x)\psi_{\alpha-1}(x+\partial_x)\psi_{\beta-1} dx 
      \\ &\qquad = \frac{1}{2\sqrt{\alpha\beta}} \int \psi_{\alpha-1} 
      (x+\partial_x)(x-\partial_x)\psi_{\beta-1} \\ &\qquad = 
      \sqrt{\frac{\beta}{\alpha}} \int \psi_{\alpha-1}\psi_{\beta-1} dx.
    \end{align*}
  </p>
  <p>
    Using the previous equation we see that
    
    \[
      \int \psi_\alpha \psi_\alpha dx = \int \psi_0^2 dx = 
      \pi^{-1/2}\int_{-\infty}^\infty e^{-x^2} dx = 1.
    \]
  </p>
  <p>
    If, on the other hand for example $\alpha\lt\beta$, then
    
    \[
      \int\psi_\alpha\psi_\beta dx = \ldots = c_{\alpha,\beta} \int 
      \psi_0\psi_{\beta-\alpha} dx = c'_{\alpha,\beta} 
      \int_{-\infty}^\infty \partial_x^{\beta-\alpha} e^{-x^2} dx = 0
    \]
    
    since all the derivatives of Gaussians vanish at infinity.
  </p>
</div>


<div class="corollary">
<span class="corollary">Corollary 2.2.</span>
<p>
  Let $\mathscr{O}=\{x, \partial_x\}$ be the set of operators containing 
  the multiplication by $x$ and differentiation by $x$ operators. Let 
  $L^0_\alpha = \{\psi_\alpha\}$ and
  
  \[
    L^{k+1}_\alpha = \{ Sf \mid S\in\mathscr{O}, \, f\in L^k_\alpha \}.
  \]
  
  Then 
  
  \[
    \norm{f}_{L^2(\R)} \leq 2^{k/2} \prod_{\ell=1}^k \sqrt{\alpha+\ell} 
    = \sqrt{ 2^k \frac{(a+k)!}{a!} }
  \]
  
  for any $f \in L^k_\alpha$.
</p>
</div>
<div class="proof">
  <span class="proof">Proof.</span>
  <p>
    The sequence $(\psi_\alpha)_\alpha$ is orthonormal by <a 
    href=#orthonormalSequenceLemma>Lemma 2.1</a>. Hence we have 
    $\norm{\psi_\alpha}_2=1$. Assume that the claim is true for 
    $L^k_\alpha$ for any $\alpha\in\N$. We will prove it for 
    $L^{k+1}_\alpha$. Let that $f\in L^k_\alpha$. Then there are 
    operators $S_1,\ldots,S_k \in \mathscr{O}$ such that $f = 
    S_1S_2\cdots S_k \psi_\alpha$.
  </p>
  <p>
    Consider the case $S_k = \partial_x$. Let $O\in\mathscr{O}$ and use 
    one of the basic identities to get
    
    \begin{align*}
      &\norm{Of}_2 = \norm{OS_1\cdots S_{k-1} \partial_x \psi_\alpha}_2 
      \\ &\qquad = \norm{OS_1\cdots S_{k-1} \left( 
      \sqrt{\tfrac{\alpha}{2}}\psi_{\alpha-1} - 
      \sqrt{\tfrac{\alpha+1}{2}}\psi_{\alpha+1}\right)}_2 \\ &\qquad 
      \leq \sqrt{\tfrac{\alpha}{2}} \norm{OS_1\cdots S_{k-1} 
      \psi_{\alpha-1}}_2 + \sqrt{\tfrac{\alpha+1}{2}} \norm{OS_1\cdots 
      S_{k-1} \psi_{\alpha+1}}_2 \\ &\qquad \leq 
      \sqrt{\tfrac{\alpha+1}{2}} 2^{k/2} \left( \prod_{\ell=1}^k 
      \sqrt{\alpha-1+\ell} + \prod_{\ell=1}^k \sqrt{\alpha+1+\ell} 
      \right) \\ &\qquad \leq 2^{(k-1)/2} \sqrt{\alpha+1} \cdot 2 
      \prod_{\ell=1}^k\sqrt{\alpha+1+\ell} \\ &\qquad \leq 2^{(k+1)/2} 
      \prod_{\ell=1}^{k+1} \sqrt{\alpha+\ell}
    \end{align*}
    
    since $OS_1\cdots S_{k-1} \psi_{\alpha-1}$ and $OS_1\cdots S_{k-1} 
    \psi_{\alpha+1}$ are in $L^k_{\alpha-1}$ and $L^k_{\alpha+1}$, 
    respectively. The same kind of deduction works in the case when 
    $S_k$ is multiplication by $x$. Hence the claim follows by 
    induction.
  </p>
</div>
</section>


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<h2>References</h2>
<h3>Offline sources</h3>
<ul class="referenceList">
  <li id="articleTry" class="reference">
    <span class="ref-nbr">[1]</span>
    <span class="ref-authors">E. Blåsten</span> and 
    <span class="ref-authors">M. Meikäläinen:</span>
    <span class="ref-title">A proof of the Riemann conjecture,</span>
    <span class="ref-journal">Annals of Mathematics,</span>
    <span class="ref-volume">23,</span>
    <span class="ref-issue">2</span>
    <span class="ref-year">(2069),</span> 
    <span class="ref-pages">1&ndash;503.</span>
  </li>
</ul>

<h3>Links</h3>
<ul class="referenceList">
  <li class="reference">
    <span class="referenceNbr">[2]</span>
    <a href="example2.html" class="addTxtForbidden">example2.html</a>
  </li>
  <li class="reference">
    <span class="referenceNbr">[3]</span>
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  </li>
</ul>
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